Closed Formulas and Determinantal Expressions for Higher-Order

Home , Euler numbers

arXiv:2005.03921v1 [math.NT] 8 May 2020 r aldcasclBrolinmesadElrnmes respectiv numbers, Euler and numbers Bernoulli classical called are npriua,tertoa numbers rational the particular, In sal endb en ftefloiggnrtn functions: generating following the of means by defined usually h lsia enul polynomials Bernoulli classical The Introduction 1 lsdfrua n eemnna expressions determinantal and formulas Closed e t te oyoil ntrso trignumbers Stirling of terms in polynomials lsdfrs eemnna expression. Determinantal forms, Closed presented. are f polynomials kind Euler second and the det Bernoulli of and order numbers formulas Stirling closed involving several expressions polynomials, Bell of erties 18,1C0 11Y55. 11C20, 11B83, − xt 1 eateto ahmtc,AdnzUniversity, Akdeniz Mathematics, of Department o ihrodrBroliadEuler and Bernoulli higher-order for nti ae,apyn h aad rn oml n oepro some and formula Bruno Faà di the applying paper, this In ahmtc ujc lsicto 2010: Classification Subject Mathematics Keywords: = X n ∞ =0 B -al [email protected] E-mail: n ( x ) enul n ue oyoil,Siln numbers, Stirling polynomials, Euler and Bernoulli n uamtChtDA Cihat Muhammet t n ! 75-nay,Turkey 07058-Antalya, ( | t | < 2 π Abstract and ) B B n 1 n = ( x B et n ue polynomials Euler and ) 2 n e 1 + 0 n integers and (0) xt = GLI ˘ X n ∞ =0 E n 16,05A19, 11B68, ( x ) n t E erminantal n ! rhigher- or n ( 2 = | t | E π < n ely. E n p- ( n x ) (1 are ) . / 2) As is well known, the classical Bernoulli and Euler polynomials play im- portant roles in different areas of mathematics such as number theory, combinatorics, special functions and analysis. Numerous generalizations of these polynomials and numbers are defined and many properties are studied in a variety of context. One of them can be (α) traced back to Nörlund [7]: The higher-order Bernoulli polynomials Bn (x) (α) and higher-order Euler polynomials En (x), each of degree n in x and in α, are defined by means of the generating functions

t α ∞ tn ext = B(α)(x) (1.1) et − 1 n n! n=0 X and 2 α ∞ tn ext = E(α)(x) , (1.2) et +1 n n! n=0 X (1) (1) respectively. For α = 1, we have Bn (x)= Bn(x) and En (x)= En(x). According to Wiki [2], ”In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations. It may contain constants, variables, certain ‘well-known’ operations (e.g.,+ − ×÷), and functions (e.g., nth root, exponent, logarithm, trigono- metric functions, and inverse hyperbolic functions), but usually no limit.” From this point of view, Wei and Qi [15] studied several closed form expressions for Euler polynomials in terms of determinant and the Stirling numbers of the second kind. Also, Qi and Chapman [8] established two closed forms for the Bernoulli polynomials and numbers involving the Stirling numbers of the second kind and in terms of a determinant of combinatorial numbers. Moreover, some special determinantal expressions and recursive formulas for Bernoulli polynomials and numbers can be found in [9]. In 2018, Hu and Kim [6] presented two closed formulas for Apostol- Bernoulli polynomials by aid of the properties of the Bell polynomials of the second kind Bn,k (x1, x2, ..., xn−k+1) (see Lemma 2.1, below). Very recently, Dai and Pan [4] have obtained the closed forms for degenerate Bernoulli polynomials. In this paper, we focus on higher-order Bernoulli and Euler polynomials in those respects, mentioned above. Firstly, we find some novel closed formulas for higher-order Bernoulli and Euler polynomials in terms of Stirling numbers of the second kind S(n, k) via the Faàdi Bruno formula for the

2 Bell polynomials of the second kind and the generating function methods. Secondly, we establish some determinantal expressions by applying a formula of higher order derivative for the ratio of two diﬀerentiable functions. Conse- quently, taking some special cases in our results provides the further formulas for classical Bernoulli and Euler polynomials, and numbers.

2 Some Lemmas

In order to prove our main results, we recall several lemmas below.

Lemma 2.1 ( [3, p. 134 and 139]) The Bell polynomials of the second kind, or say, partial Bell polynomials, denoted by Bn,k (x1, x2, ..., xn−k+1) for n ≥ k ≥ 0, deﬁned by

∞ l−k+1 li n! xi Bn,k (x1, x2, ..., xn−k+1)= l−k+1 . l ! i=1 i! 1≤i≤n, li∈{0}∪N i=1 i n X n Q i=1ili=n, i=1li=k Q The Fa`adi Bruno formula canP be describedP in terms of the Bell polynomials of the second kind Bn,k (x1, x2, ..., xn−k+1) by

n dn f ◦ h (t)= f (k) (h (t)) B h′ (t) , h′′ (t) , ..., h(n−k+1) (t) . (2.1) dtn n,k k=0 X Lemma 2.2 ( [3, p. 135]) For n ≥ k ≥ 0, we have

2 n−k+1 k n Bn,k abx1, ab x2, ..., ab xn−k+1 = a b Bn,k (x1, x2, ..., xn−k+1) , (2.2) where a and b are any complex number.

Lemma 2.3 ( [3, p. 135])For n ≥ k ≥ 0, we have

Bn,k (1, 1, ..., 1) = S (n, k) , (2.3) where S (n, k) is the Stirling numbers of the second kind, deﬁned by [3, p. 206] (et − 1)k ∞ tn = S (n, k) . k! n! n k X= 3 Lemma 2.4 ( [5])For n ≥ k ≥ 1, we have

k 1 1 1 n! n + k B , , ..., = (−1)k−i S (n + i, i) . n,k 2 3 n − k +2 (n + k)! k − i i=0 X (2.4)

Lemma 2.5 ([1, p. 40, Entry 5]) For two diﬀerentiable functions p (x) and q (x) =06 , we have for k ≥ 0

dk p (x) dxk q (x) p q 0 ... 0 0 p′ q′ q . . . 0 0 ′′ ′′ 2 ′ p q q ... 0 0 1 k ...... (−1) = ...... (2.5) qk+1 ...... (k−2) (k−2) k−2 (k−3) p q 1 q ... q 0 k k p(k−1) q(k−1) −1 q(k−2) ... −1 q′ q 1 k−2 p(k) q(k) k q(k−1) ... k q′′ k q′ 1 k−2 k−1

In other words, the formula (2.5 ) can be represented as

dk p (x) (−1)k = W (x) , dxk q (x) qk+1 (k+1)×(k+1)

where W(k+1)×(k+1) (x) denotes the determinant of the matrix

W(k+1)×(k +1) (x)= U(k+1)×1 (x) V(k+1)×k (x) .

(l−1) Here U(k+1)×1 (x) has the elements ul,1 (x)= p (x) for 1 ≤ l ≤ k +1 and V(k+1)×k (x) has the entries of the form

i−1 (i−j) j−1 q (x) , if i − j ≥ 0; vi,j (x)= (0, if i − j < 0, for 1 ≤ i ≤ k +1 and 1 ≤ j ≤ k.

4 3 Closed formulas

In this section, we give closed formulas for higher-order Bernoulli and Euler polynomials.

(α) Theorem 3.1 The higher-order Bernoulli polynomials Bn (x) can be ex- pressed as

n k i n k! k + i B(α)(x)= h−αi (−1)i−j S(k + j, j)xn−k, n k i (k + i)! i − j k i=0 j=0 X=0 X X where S(n, k) is the Stirling numbers of the second kind and hxin denotes the falling factorial, deﬁned for x ∈ R by

n −1 x (x − 1) ...(x − n + 1), if n ≥ 1; hxin = (x − k)= k (1, if n =0. Y=0 (α) In particular x = 0, the higher-order Bernoulli numbers Bn possess the following form

n i n! n + i B(α) = h−αi (−1)i−j S(n + j, j). (3.1) n i (n + i)! i − j i=0 j=0 X X et − 1 α e α Proof. Let us begin by writing = st−1ds . From (2.1) and t 1 (2.4), we have Z

dk et − 1 −α dtk t k et − 1 −α−i e e e = h−αi B st−1 ln sds, st−1 ln2 sds, ..., st−1 lnk−i+1 sds i t k,i i=0 Z1 Z1 Z1 Xk 1 1 1 → h−αi B , , ..., , as t → 0 i k,i 2 3 n − i +2 i=0 Xk i k! k + i = h−αi (−1)i−j S (k + j, j) , i (k + i)! i − j i=0 j=0 X X 5 Also (ext)(k) = xkext → xk, as t → 0. So, using the Leibnitz’s formula for the nth derivative of the product of two functions yields that

dn et − 1 −α lim ext t→0 dtn t " # n k i n k! k + i = h−αi (−1)i−j S (k + j, j) xn−k, k i (k + i)! i − j k i=0 j=0 X=0 X X (α) which is equal to Bn (x) from the generating function (1.1). For x = 0, we immediately get the identity (3.1).

i Remark 3.2 For α =1, noting the fact h−1ii =(−1) i!, the equation (3.1) becomes n i n! n + i B = i! (−1)j S(n + j, j) n (n + i)! i − j i=0 j=0 Xn X n S(n + j, j) i = (−1)j n+j j j=0 j i=j X X n n+1 j j+1 = (−1) n+j S(n + j, j), j=0 j X which is [8, Eq. 1.3].

(α) Theorem 3.3 The higher-order Euler polynomials En (x) can be represented as n k n S(k, i) E(α)(x)= h−αi xn−k. n k i 2i k i=0 X=0 X (α) In particular, the higher-order Euler numbers En have the following form

n k n E(α) = h−αi S(k, i)2k−i. (3.2) n k i k i=0 X=0 X Proof. By (2.1), (2.2) and (2.3), we have

k dk et +1 −α et +1 −α−i et et et = h−αi B , , ..., dtk 2 i 2 k,i 2 2 2 i=0 X 6 k et +1 −α−i et i = h−αi B (1, 1, ..., 1) i 2 2 k,i i=0 Xk S (k, i) → h−αi , as t → 0. i 2i i=0 X So, from the Leibnitz’s rule again and the generating function for higher- order Euler polynomials, given by (1.2), we obtain that

n t −α (α) d e +1 xt En (x) = lim e t→0 dtn 2 " # n k n S (k, i) = h−αi xn−k. k i 2i k i=0 X=0 X Taking special case gives the closed form (3.2) immediately for higher-order Euler numbers.

Remark 3.4 For α = 1, the counterpart closed formula for Euler polynomials can be derived. Moreover, setting more special cases leads to similar formula for Euler numbers.

4 Determinantal expressions

This section is devoted to demonstrate some determinantal expressions for higher order Bernoulli and Euler polynomials.

(α) Theorem 4.1 The higher-order Bernoulli polynomials Bn (x) can be represented in terms of the following determinant as

1 γ0 0 ... 0 0 x γ γ ... 0 0 1 0 x2 γ 2 γ ... 0 0 2 1 1 (α) n ...... Bn (x)=(−1) . . . . . . , n−2 n−2 x γn−2 1 γn−3 ... γ0 0 n−1 n−1 n−1 x γn−1 1 γn−2 ... n−2 γ1 γ0 n n n n x γn γn−1 ... γ2 γ1 1 n−2 n−1

7 where n i n! n + i γ = hαi (−1)i−j S (n + j, j) . n i (n + i)! i − j i=0 j=0 X X Proof. We use Lemma 2.5 for p (t) = ext and q (t) = ((et − 1) /t)α . Note that if we proceed similar manipulations to the proof of Theorem 3.1, then, we deduce that

n n i d n! i−j n + i lim q (t)= hαii (−1) S (n + j, j) := γn. t→0 dtn (n + i)! i − j i=0 j=0 X X So, we have dn ext dtn ((et − 1) /t)α ext q 0 ... 0 0 xext q′ q . . . 0 0 2 xt ′′ 2 ′ n x e q 1 q ... 0 0 (−1) ...... = ...... t (n+1)α ((e − 1) /t) n−2 xt (n−2) n−2 (n−3) x e q 1 q ... q 0 n−1 xt (n−1) n−1 (n−2) n−1 ′ x e q 1 q ... n−2 q q n n n xnext q(n) q(n−1) ... q′′ q′ 1 n−2 n−1

1 γ0 0 ... 0 0 x γ γ ... 0 0 1 0 x2 γ 2 γ ... 0 0 2 1 1 n ...... → (−1) . . . . . . n−2 n−2 x γn−2 1 γn−3 ... γ0 0 n−1 n−1 n−1 x γn−1 1 γn−2 ... n−2 γ1 γ0 n n n n x γn γn−1 ... γ2 γ1 1 n−2 n−1

as t → 0. From the generating function, given by (1.1), we reach the desired result.

Remark 4.2 We mention that for the special case x =0, the analog deter- (α) minantal expression for higher-order Bernoulli numbers Bn can be oﬀered. Moreover, for α = 1, and α = 1 and x = 0, the similar representations can be obtained for classical Bernoulli polynomials and numbers, respectively.

8 (α) Theorem 4.3 The higher-order Euler polynomials En (x) can be represented in terms of the following determinant as

1 β0 0 ... 0 0 x β β ... 0 0 1 0 x2 β 2 β ... 0 0 2 1 1 (α) n ...... En (x)=(−1) . . . . . . , n−2 n−2 x βn−2 1 βn−3 ... β0 0 n−1 n−1 n−1 x βn−1 1 βn−2 ... n−2 β1 β0 n n n n x βn βn−1 ... β2 β1 1 n−2 n−1 where n S (n, i) β = hαi . n i 2i i=0 X Proof. The proof can be veriﬁed by proceeding as in the proof of Theorem 4.1. So, we omit it.

Remark 4.4 The special values of the Bell polynomials of the second kind Bn,k are worthy in combinatorics and number theory. In this respect, Bn,k has been applied in order to cope with some diﬃcult problems and obtain signiﬁcant results in many studies (see for example [10–14]).

References

[1] N. Bourbaki, Functions of a Real Variable, Elementary Theory, Trans- lated from the 1976 French Original by Philip Spain. Elements of Math- ematics (Berlin). Springer, Berlin, 2004.

[2] Closed-form expression. https://en.wikipedia.org/wiki/Closed- form expression.

[3] L. Comtet, Advanced Combinatorics: The Art of Finite and Inﬁnite Expansions, Revised and Enlarged Edition. D. Reidel Publishing Co., Dordrecht 1974.

[4] L. Dai and H. Pan, Closed forms for degenerate Bernoulli polynomials, Bull. Aust. Math. Soc. 101 (2020), no. 2, 207–217.

9 [5] B.-N. Guo and F. Qi, An explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind, J. Anal. Number Theory 3 (2015), no. 1, 27–30.

[6] S. Hu and M.-S. Kim, Two closed forms for the Apostol–Bernoulli polynomials, Ramanujan J. 46 (2018), no. 1, 103–117.

[7] N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924.

[8] F. Qi and R. J. Chapman, Two closed forms for the Bernoulli polynomials, J. Number Theory, 159 (2016), 89–100.

[9] F. Qi and B.-N. Guo, Some Determinantal Expressions and Recurrence Relations of the Bernoulli Polynomials, Mathematics, 4 (2016), no. 4, 1–11.

[10] F. Qi, Derivatives of tangent function and tangent numbers, Appl. Math. Comput. 268 (2015), 844–858.

[11] F. Qi and M.-M. Zheng, Explicit expressions for a family of the Bell polynomials and applications, Appl. Math. Comput. 258 (2015), 597– 607.

[12] F. Qi and B.-N. Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterr. J. Math. 14 (2017), no. 3, Article 140, 14 pages.

[13] F. Qi, D. Lim, and B.-N. Guo, Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to diﬀerential equations, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RAC- SAM, 113 (2019), 1–9.

[14] F. Qi, An Explicit Formula for the Bell Numbers in Terms of the Lah and Stirling Numbers, Mediterr. J. Math. 13 (2016), 2795–2800.

[15] C.-F. Wei and F. Qi, Several closed expressions for the Euler numbers, J. Inequal. Appl. 2015, 219 (2015).